OFFSET
1,2
COMMENTS
Also number of bracelets (or necklaces) with n red or blue beads such that the beads switch colors when bracelet is turned over.
a(n) is also the number of frieze patterns generated by filling a 1 X n block with n copies of an asymmetric motif (where the copies are chosen from original motif or a 180-degree rotated copy) and then repeating the block by translation to produce an infinite frieze pattern. (Pisanski et al.)
a(n) is also the number of minimal fibrations of a bidirectional n-cycle over the 2-bouquet up to precompositions with automorphisms of the n-cycle. (Boldi et al.) - Sebastiano Vigna, Jan 08 2018
REFERENCES
Jeb F. Willenbring, A stability result for a Hilbert series of O_n(C) invariants.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..3334
Rémi Cocou Avohou, Joseph Ben Geloun, and Reiko Toriumi, Counting U(N)^{⊗r} ⊗ O(N)^{⊗q} invariants and tensor model observables, Eur. Phys. J. C 84, 839 (2024), see pp. 11, 27; Preprint arXiv:2404.16404 [hep-th], 2024. See pp. 18, 49.
Paolo Boldi and Sebastiano Vigna, Fibrations of Graphs, Discrete Math., 243 (2002), 21-66.
Shinsaku Fujita, alpha-beta Itemized Enumeration of Inositol Derivatives and m-Gonal Homologs by Extending Fujita's Proligand Method, Bull. Chem. Soc. Jpn. 2017, 90, 343-366. See Table 8.
T. Pisanski, D. Schattschneider, and B. Servatius, Applying Burnside's lemma to a one-dimensional Escher problem, Math. Mag., 79 (2006), 167-180.
Jeb F. Willenbring, Home page.
A. Yajima, How to calculate the number of stereoisomers of inositol-homologs, Bull. Chem. Soc. Jpn. 2014, 87, 1260-1264. See Tables 1 and 2 (and text).
FORMULA
G.f.: x/(1-x) + x^2/(2*(1-2*x^2)) + Sum_{n >= 1} (x^(2*n)/(2*n)) * Sum_{ d divides n } phi(d)/(1-x^d)^(2*n/d), or x^2/(2*(1-2*x^2)) - Sum_{n >= 1} phi(n)*log(1-2*x^n)/(2*n). [corrected and extended by Andrey Zabolotskiy, Oct 17 2017]
a(n) = A000031(n)/2 + (if n even) 2^(n/2-2).
EXAMPLE
2 at n=3 because there are two such cycles. On (o -> o -> o ->) and (o -> o <- o ->).
MAPLE
v:=proc(n) local k, t1; t1:=0; for k in divisors(n) do t1 := t1+phi(k)*2^(n/k); od: t1; end;
h:=n-> if n mod 2 = 0 then (n/2)*2^(n/2); else 0; fi;
A053656:=n->(v(n)+h(n))/(2*n); # N. J. A. Sloane, Nov 11 2006
MATHEMATICA
a[n_] := Total[ EulerPhi[#]*2^(n/#)& /@ Divisors[n]]/(2n) + 2^(n/2-2)(1-Mod[n, 2]); Table[a[n], {n, 1, 35}] (* Jean-Fran�ois Alcover, Nov 21 2011 *)
PROG
(PARI) a(n)={(sumdiv(n, d, eulerphi(d)*2^(n/d))/n + if(n%2==0, 2^(n/2-1)))/2} \\ Andrew Howroyd, Jun 16 2021
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
Jeb F. Willenbring (jwillenb(AT)ucsd.edu), Feb 14 2000
EXTENSIONS
More terms and additional comments from Christian G. Bower, Dec 13 2001
STATUS
approved